An Information Theoretic Framework for Eukaryotic Gradient Sensing

Let the random variables {Bj} N j=1 represent the states of theN cAMP receptors on the cell surface; Bj = 1 if the receptor is bound to a molecule of cAMP, otherwise Bj = 0. Each receptor is taken to lie on the surface of a sphere of radius R = 7.5μm, with displacement from the center of the cell in a direction given by a unit vector ~x ∈ S. We set the center of the sphere to be the origin of our coordinate system. Invoking the assumption of gradient linearity, we take the equilibrium concentration of cAMP at ~x to be c(~x|~g) = a+ b(~x ·~g) where ~g ∈ S is a unit vector in the direction of the gradient. The parameter a is the mean concentration over the cell surface, and b = R|~ ∇c| is half the drop in concentration from one extreme on the cell surface to the other. Before the stimulus occurs, the gradient direction is undefined.